Dirichlet problem and subclasses of Baire-one functions

作者：Posta Petr^*

来源：Israel Journal of Mathematics, 2018, 226(1): 177-188.

DOI：10.1007/s11856-018-1707-z

Let U be a bounded open subset of a"e (d) , d 2 and f a C(a,U). The Dirichlet solution f (CU) of the Dirichlet problem associated with the Laplace equation with a boundary condition f is not continuous on the closure A> of U in general if U is not regular but it is always Baire-one.
Let H(U) be the space of all functions continuous on the closure A> and harmonic on U and F(H(U)) be the space of uniformly bounded absolutely convergent series of functions in H(U). We prove that f (CU) can be obtained as a uniform limit of a sequence of functions in F(H(U)). Thus f (CU) belongs to the subclass B (1/2) of Baire-one functions studied for example in [8]. This is not only an improvement of the result obtained in [10] but it also shows that the Dirichlet solution on the closure A> can share better properties than to be only a Baire-one function. Moreover, our proof is more elementary than that in [10].
A generalization to the abstract context of simplicial function space on a metrizable compact space is provided.
We conclude the paper with a brief discussion on the solvability of the abstract Dirichlet problem with a boundary condition belonging to the space of differences of bounded semicontinuous functions complementing the results obtained in [17].

• 出版日期2018-6